Theory question 109
Differentiation(a) Find the derivative of 4x − 7/x2 with respect to x, from first principle.
(b) Given that tan P = 3/(x − 1) and tan Q = 2/(x − 1), find tan (P − Q).
Show the model answer
Model answer
(a) y + δy = 4(x + δx) − 7/(x + δx)²; δy = 4δx + 7[1/x² − 1/(x + δx)²] = 4δx + 7(2xδx + δx²)/(x²(x + δx)²).
δy/δx = 4 + 7(2x + δx)/(x²(x + δx)²). As δx → 0: dy/dx = 4 + 14x/x⁴ = 4 + 14/x³.
(b) tan(P − Q) = (3/(x − 1) − 2/(x − 1)) ÷ (1 + 6/(x − 1)²) = (1/(x − 1)) × (x − 1)²/((x − 1)² + 6) = (x − 1)/(x² − 2x + 7).
Explanation
From first principles: find δy, divide by δx, then let δx → 0.